opytimark.markers.many_dimensional

class opytimark.markers.many_dimensional.BiggsExponential3(name: Optional[str] = 'BiggsExponential3', dims: Optional[int] = 3, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

BiggsExponential3 class implements the Biggs Exponential’s 3rd benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3) = \sum_{i=1}^{10}(e^{-t_ix_1} - x_3e^{-t_ix_2} - y_i)^2\]

Domain:

The function is commonly evaluated using \(x_i \in [0, 20] \mid i = \{1, 2, 3\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 10, 5)\).

class opytimark.markers.many_dimensional.BiggsExponential4(name: Optional[str] = 'BiggsExponential4', dims: Optional[int] = 4, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

BiggsExponential4 class implements the Biggs Exponential’s 4th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3, x_4) = \sum_{i=1}^{10}(x_3e^{-t_ix_1} - x_4e^{-t_ix_2} - y_i)^2\]

Domain:

The function is commonly evaluated using \(x_i \in [0, 20] \mid i = \{1, 2, 3, 4\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 10, 1, 5)\).

class opytimark.markers.many_dimensional.BiggsExponential5(name: Optional[str] = 'BiggsExponential5', dims: Optional[int] = 5, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

BiggsExponential5 class implements the Biggs Exponential’s 5th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3, x_4, x_5) = \sum_{i=1}^{11}(x_3e^{-t_ix_1} - x_4e^{-t_ix_2} + 3e^{-t_ix_5} - y_i)^2\]

Domain:

The function is commonly evaluated using \(x_i \in [0, 20] \mid i = \{1, 2, 3, 4, 5\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 10, 1, 5, 4)\).

class opytimark.markers.many_dimensional.BiggsExponential6(name: Optional[str] = 'BiggsExponential6', dims: Optional[int] = 6, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

BiggsExponential6 class implements the Biggs Exponential’s 6th benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3, x_4, x_5, x_6) = \sum_{i=1}^{13}(x_3e^{-t_ix_1} - x_4e^{-t_ix_2} + x_6e^{-t_ix_5} - y_i)^2\]

Domain:

The function is commonly evaluated using \(x_i \in [0, 20] \mid i = \{1, 2, 3, 4, 5, 6\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 10, 1, 5, 4, 3)\).

class opytimark.markers.many_dimensional.BoxBetts(name: Optional[str] = 'BoxBetts', dims: Optional[int] = 3, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

BoxBetts class implements the BoxBetts’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3) = \sum_{i=1}^{n}g(x)^2\]

\[g(x) = e^{-0.1(i+1)x_1} - e^{-0.1(i+1)x_2} - (e^{-0.1(i+1)} - e^{-(i+1)}*x_3)\]

Domain:

The function is commonly evaluated using \(x_1 \in [0.9, 1.2], x_2 \in [9, 11.2], x_3 \in [0.9, 1.2]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 10, 1)\).

class opytimark.markers.many_dimensional.Colville(name: Optional[str] = 'Colville', dims: Optional[int] = 4, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Colville class implements the Colville’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3, x_4) = 100(x_1 - x_2^2)^2 + (1 - x_1)^2 + 90(x_4 - x_3^2)^2 + (1 - x_3)^2 + 10.1((x_2-1)^2 + (x_4-1)^2) + 19.8(x_2-1)(x_4-1)\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, 3, 4\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 1, 1, 1)\).

class opytimark.markers.many_dimensional.GulfResearch(name: Optional[str] = 'GulfResearch', dims: Optional[int] = 3, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

GulfResearch class implements the GulfResearch’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3) = \sum_{i=1}^{99}[e^(-\frac{(u_i-x_2)^x_3}{x_1}) - 0.01i]^2\]

Domain:

The function is commonly evaluated using \(x_1 \in [0.1, 100], x_2 \in [0, 25.6], x_3 \in [0, 5]\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (50, 25, 1.5)\).

class opytimark.markers.many_dimensional.HelicalValley(name: Optional[str] = 'HelicalValley', dims: Optional[int] = 3, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

HelicalValley class implements the Helical Valley’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3) = 100[(x_3-10\theta)^2 + (\sqrt{x_1^2+x_2^2}-1)] + x_3^2\]

Domain:

The function is commonly evaluated using \(x_i \in [-10, 10] \mid i = \{1, 2, 3\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (1, 0, 0)\).

class opytimark.markers.many_dimensional.MieleCantrell(name: Optional[str] = 'MieleCantrell', dims: Optional[int] = 4, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

MieleCantrell class implements the MieleCantrell’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3, x_4) = (e^{-x_1} - x_2)^4 + 100(x_2 - x_3)^6 + (tan(x_3-x_4))^4 + x_1^8\]

Domain:

The function is commonly evaluated using \(x_i \in [-1, 1] \mid i = \{1, 2, 3, 4\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 1, 1, 1)\).

class opytimark.markers.many_dimensional.Paviani(name: Optional[str] = 'Paviani', dims: Optional[int] = 10, continuous: Optional[bool] = True, convex: Optional[bool] = True, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Paviani class implements the Paviani’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, \ldots, x_10) = \sum_{i=1}^{10}[ln(x_i-2)^2 + ln(10-x_i)^2] - (\prod_{i=1}^{10}x_i)^{0.2}\]

Domain:

The function is commonly evaluated using \(x_i \in [2.0001, 10] \mid i = \{1, 2, \ldots, 10\}\).

Global Minima:

\(f(\mathbf{x^*}) = -45.778452053828865 \mid \mathbf{x^*} = (9.351, 9.351, \ldots, 9.351)\).

class opytimark.markers.many_dimensional.SchmidtVetters(name: Optional[str] = 'SchmidtVetters', dims: Optional[int] = 3, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

SchmidtVetters class implements the Schmidt Vetters’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3) = \frac{1}{1 + (x_1-x_2)^2} + sin(\frac{\pi x_2+x_3}{2}) + e^{(\frac{x_1+x_2}{x_2}-2)^2}\]

Domain:

The function is commonly evaluated using \(x_i \in [0, 2] \mid i = \{1, 2, 3\}\).

Global Minima:

\(f(\mathbf{x^*}) = 3 \mid \mathbf{x^*} = (0.78547, 0.78547, 0.78547)\).

class opytimark.markers.many_dimensional.Simpleton(name: Optional[str] = 'Simpleton', dims: Optional[int] = 10, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Simpleton class implements the Simpleton’s Problem benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3, x_4, x_5, x_6, x_7, x_8, x_9, x_10) = \frac{x_1 x_2 x_3 x_4 x_5}{x_6 x_7 x_8 x_9 x_10}\]

Domain:

The function is commonly evaluated using \(x_i \in [1, 10] \mid i = \{1, 2, \ldots, 10\}\).

Global Minima:

\(f(\mathbf{x^*}) = 10^5 \mid \mathbf{x^*} = (10, 10, 10, 10, 10, 1, 1, 1, 1, 1)\).

class opytimark.markers.many_dimensional.Watson(name: Optional[str] = 'Watson', dims: Optional[int] = 6, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = False)

Bases: opytimark.core.Benchmark

Watson class implements the Watson’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3, x_4, x_5, x_6) = \sum_{i=0}^{29}{\sum_{j=0}^{4}((j-1)a_i^j x_{j+1}) - [\sum_{j=0}^{5}a_i^j x_{j+1}]^2 - 1}^2 + x_1^2\]

Domain:

The function is commonly evaluated using \(|x_i| \leq 10 \mid i = \{1, 2, 3, 4, 5, 6\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0.002288 \mid \mathbf{x^*} = (−0.0158, 1.012, −0.2329, 1.260, −1.513, 0.9928)\).

class opytimark.markers.many_dimensional.Wolfe(name: Optional[str] = 'Wolfe', dims: Optional[int] = 3, continuous: Optional[bool] = True, convex: Optional[bool] = False, differentiable: Optional[bool] = True, multimodal: Optional[bool] = True, separable: Optional[bool] = True)

Bases: opytimark.core.Benchmark

Wolfe class implements the Wolfe’s benchmarking function.

\[f(\mathbf{x}) = f(x_1, x_2, x_3) = \frac{4}{3}(x_1^2 + x_2^2 - x_1x_2)^{0.75} + x_3\]

Domain:

The function is commonly evaluated using \(x_i \in [0, 2] \mid i = \{1, 2, 3\}\).

Global Minima:

\(f(\mathbf{x^*}) = 0 \mid \mathbf{x^*} = (0, 0, 0)\).